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A class of interacting particle systems on $\mathbb{Z}$, involvinginstantaneously annihilating or coalescing nearest neighbour random walks, areshown to be Pfaffan point processes for all deterministic initial conditions.As diffusion limits, explicit Pfaffan kernels are derived for a variety ofcoalescing and annihilating Brownian systems. For Brownian motions on$\mathbb{R}$, depending on the initial conditions, the corresponding kernelsare closely related to the bulk and edge scaling limits of the Pfaffan pointprocess for real eigenvalues for the real Ginibre ensemble of random matrices.For Brownian motions on $\mathbb{R}_{+}$ with absorbing or reflected boundaryconditions at zero new interesting Pfaffan kernels appear. We illustrate theutility of the Pfaffan structure by determining the extreme statistics of therightmost particle for the purely annihilating Brownian motions, and alsocomputing the probability of overcrowded regions for all models.

Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: annihilating and coalescing random walks